The coefficient of mutual inductance

Explanation

Consider a small strip of area `dA=adx` Magnetic field at distance `x` from long wire
`B=(mu_(0)i)/(2pix)`
`dphi=BdA=(mu_(0)ia)/(2pi)(dx)/(x)`
`phi=(mu_(0)ia)/(2pi)int_(0)^(b+a)(dx)/(x)`
`=(mu_(0)ia)/(2pi)|1nx|_(b)^(b+a)`
`=(mu_(0)ia)/(2pi)[1n(b+a)-1n(b)]`
`=(mu_(0)ai)/(2pi)1n(1+(a)/(b))`
`phi=Mi`
`M=(mu_(0)a)/(2pi)1n(1+(a)/(b))`